Approximation Algorithms for $\ell_p$-Shortest Path and $\ell_p$-Group Steiner Tree

TL;DR

This paper develops polylogarithmic approximation algorithms for vector-cost variants of shortest path and Steiner problems, using new flow-based relaxations, with results applicable to both series-parallel and general graphs.

Contribution

It introduces novel flow-based Sum-of-Squares relaxations for vector-cost problems and provides approximation algorithms with polylogarithmic guarantees.

Findings

Algorithms run in polynomial time for series-parallel graphs.

Algorithms run in quasi-polynomial time for arbitrary graphs.

Includes hardness results indicating limits of approximation.

Abstract

We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a non-negative vector cost $c_e \in \mathbb{R}^{\ell}_{\ge 0}$. For a feasible solution - a path, subtree, or tour (respectively) - we find the total vector cost of all the edges in the solution and then compute the $\ell_p$-norm of the obtained cost vector (we assume that $p \ge 1$ is an integer). Our algorithms for series-parallel graphs run in polynomial time and those for arbitrary graphs run in quasi-polynomial time. To obtain our results, we introduce and use new flow-based Sum-of-Squares relaxations. We also obtain a number of hardness results.